33. s영역에서의 회로해석-마디, 메쉬해석, 중첩원리, 전원변환, 테브난, 노턴 이론
s영역에서의 메쉬해석
4s+2=3sI1+10(I1−I2), 2s+1=10(I2−I1)+4sI2
4s+2=(10+3s)I1−10I2, 2s+1=−10I1+(4s+10)I2
I1=2s(4s2+19s+20)20s4+66s3+73s2+57s+30, I2=30s2+43s+6(s+2)(20s3+26s2+21s+15)
i1(t)=L−1{I1(s)}=−96.39e−2t−344.8e−t+841.2e−0.15tcos0.8529t+199.7e−0.15tsin0.8529tmA i2(t)=L−1{I2(t)}=−481.9e−2t−241.4e−t+723.3e−0.15tcos0.8529t+472e−0.15tsin0.8529tmA
s영역에서의 마디해석(vC(0−)=2V)
−1=Vx−7/s2/s+Vx+Vx−4/s4s
Vx=10s2+4s(2s2+4s+1)=5s2+2s(s+1+√2/2)(s+1−√2/2)
vx(t)=L−1{Vx}=4+6.864e−1.707t−5.864e−0.2929tV
VC=7s−Vx=4s2+28s+3s(2s2+4s+1), vC(0+)=lim.
s영역에서의 전원변환
\displaystyle\mathbf{Z}_{1}=\frac{2}{\mathbf{s}}||10=\frac{20}{10\mathbf{s}+2}
\displaystyle\mathbf{Z}_{2}=\mathbf{Z}_{1}+\frac{2}{\mathbf{s}}=\frac{40\mathbf{s}+4}{\mathbf{s}(10\mathbf{s}+2)}
\displaystyle\mathbf{V}_{2}(\mathbf{s})=\frac{\mathbf{s}^{2}}{\mathbf{s}^{2}+9}\frac{20}{10\mathbf{s}+2}
\displaystyle\mathbf{V}(\mathbf{s})\frac{9\mathbf{s}}{9\mathbf{s}+(40\mathbf{s}+4)/\mathbf{s}(10\mathbf{s}+2)}=\frac{180\mathbf{s}^{4}}{(\mathbf{s}^{2}+9)(90\mathbf{s}^{3}+18\mathbf{s}^{2}+40\mathbf{s}+4)}
\displaystyle v(t)=\mathcal{L}^{-1}\{\mathbf{V}(\mathbf{s})\}=5.590\times10^{-5}e^{-0.1023t}+2.098\cos(3t+3.912^{\circ})+0.1017e^{-0.04885t}\cos(0.6573t+157.9^{\circ})\text{V}.
s영역에서의 테브난 등가회로
입력 임피던스(1\text{A}의 시험전원 연결): \mathbf{Z}_{in}=\frac{\mathbf{V}_{\text{in}}}{1\text{A}}=\mathbf{V}_{\text{in}}
\displaystyle1+g\mathbf{V}_{\pi}=\frac{\mathbf{V}_{\text{in}}}{\mathbf{Z}_{\text{eq}}}, \displaystyle\mathbf{Z}_{eq}=R_{E}||\frac{1}{\mathbf{s}}C_{\pi}||r_{\pi}=\frac{R_{E}r_{\pi}}{r_{\pi}+R_{E}+\mathbf{s}R_{E}r_{\pi}C_{\pi}}
\mathbf{V}_{\pi}=-\mathbf{V}_{\text{in}}, \mathbf{Z}_{\text{in}}=\mathbf{V}_{\text{in}}, \displaystyle1-g\mathbf{V}_{\text{in}}=\frac{\mathbf{V}_{\text{in}}}{\mathbf{Z}_{\text{eq}}}, \displaystyle\mathbf{V}_{\text{in}}=\frac{\mathbf{Z}_{eq}}{1+g\mathbf{Z}_{eq}}=\frac{1}{1/\mathbf{Z}_{eq}+g}, \displaystyle\mathbf{Z}_{\text{in}}=\frac{R_{E}r_{\pi}}{r_{\pi}+R_{E}+\mathbf{s}R_{E}r_{\pi}C_{\pi}+gR_{E}r_{\pi}}.
참고자료:
Engineering Circuit Analysis 8th edition, Hayt, Kemmerly, Durbin, McGraw-Hill
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